Luneberg lens with reflecting band located at internal focus



ul- 3l. 1965 MARIE-PIERRE PRACHE 3304.244

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Filed Jun 25. 1963 4 Shoots-Shut l.

Aug. 31, 1965 MARIE-PIERRE PRAcHE 3,204,244

LUNEBERG LENS 'ITB RBFLECTING BAND LOCTED AT INTERNAL FOCUS Filed June25. 1963 4 Sheets-Sheet 2 1719.4 F'lg.5

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AUS- 31, 1965 MARIE-PIERRE PRACHE 3,204,244

LUNEBERG LENS '1TH RBFLECTING BAND LOCATED AT INTERNAL FOCUS Filed June25. 1963 4 Smets-Sheet 3 A118- 31, 1965 MARIE-PIERRE PRAcHE 3,204,244

LUNBBERG LENS 'ITB REFLEGTING BAND LOCATED AT INTERNAL FOCUS Filed June25. 1963 4 Sheets-Sheet 4 as lm Fig.1o

United States Patent O 6 claims. (ci. 343-911) The present inventionrelates to spherical reflectors for high-frequency electro-magneticwaves, and its principal object consists in sending back towards thewave source as large a quantity of energy as possible when the source isin any direction which forms with a pre-determined equatorial plane,known as the reference plane, an angle of inclination less than a givenangle, known as the limiting angle.

These reflectors are employed for example on marine beacons; in thiscase, the direction of the incident rays coming from the source ispractically horizontal, and the limiting angle is determined by themaximum inclination which the beacon, and in consequence the referenceplane, can take under the action of the sea and wind. These reflectorsmay also be utilized in aeronautical beacons, which receive wavesinclined to the horizontal plane.

The reflectors to which the invention relates comprise a sphere ofdielectric material, the permittivity of which varies in accordance witha pre-determined law as a function of the distance to the centre of thesphere.

Rellectors are already known which comprise a sphere of this kindprovided with a metallized dome or cap, but they only reflect the rayswhich come from a cone having an opening or apex angle which cannotexceed 135 if the whole of the energy is to be reflected. It has alsobeen proposed to produce a reflector which sends back the energy towardsa source located in any direction which forms with the reference planean angle less than the limiting angle, by metallizing on the outersurface of the sphere a zone which is limited by two lines parallel tothe said reference plane or equatorial plane of the sphere, the twoparallel lines limiting the said zone being located on each side of thereference plane. In the description which follows below, a sphericalzone of this kind will be termed a belt in order to simplify thedescription.

A reflector of this kind with a belt on its outer surface has a usefulsurface area which decreases rapidly with the angular opening of thebelt and the angle of inclination,A the useful surface being defined asthe surface measured in a plane perpendicular to the incident rays, andin which the rays can be sent back to the source after a singlereflection, without being stopped by the belt.

It is furthermore known that due to diffraction, the maximum gain,expressed as a ratio of powers, of a spherical reflector having agreat-circle surface S, cannot exceed 41rS/x2, in which x is thewave-length. The maximum equivalent surface is therefore 41rS2/x2. Itcan be assumed without considerable error that'the incident Wave passesthrough the reflector without diffraction, and that this diffractiononly takes place after passing through the rellector. Under theseconditions, the equivalent surface area of a reflector with a belt is41rSu2/x2, in which Su is the useful surface area of this latterreflector.

One of the objects of the present invention is to increase the usefulsurface area irrespective of the angular opening of the belt andregardless of the angle of inclination. For this purpose, the sphere hasbeen provided with an internal belt instead of an external belt.

According to the invention, a reflector for high-frequencyelectro-magnetic waves comprises a sphere of di- 3,204,244 Patented Aug.31, 1965 p ICC electric material, the permittivity of which is the sameat all points located at the same distance from the centre of saidsphere and varies as a function of the distance to the centre of thesphere in accordance with a law such that the image of the point atinfinity is located practically at a distance from the centre of thesphere which is less than the radius of the sphere, and in which a beltopaque to the electro-magnetic waves is placed at the said distance fromthe centre. This opaque belt may be formed by metallizing part of theouter surface of the inner sphere which has for its radius the saiddistance from the centre.

In accordance with one form of embodiment of the invention, the sphereof dielectric material is made-up of a number of superimposed sphericallayers, each of which has a dielectric constant different from that ofthe other layers, and the belt can then be formed by metallization of aportion of the outer surface of an internal layer.

The particular features and advantages of the invention will be betterunderstood by means of the description which follows below of someexamples of construction, given by way of explanation and without anylimitative sense, reference being made to the accompanying drawings, inwhich:

FIG. 1 represents a cross-section of the reflector according to theinvention, taken in a plane passing through the diameter perpendicularto the reference plane.

FIG. 2 shows a reflector with superimposed layers.

FIG. 3 show the trajectory of a ray.

FIGS. 4 and 5 show respectively a front view and a profile view incross-section of the reflector.

FIG. 6 shows the upper plane of the belt.

FIGS. 7, 8 and 9 show the useful surface of the reflector for differentcases.

FIG. 10 shows the equivalent surface loss with respect to a reflectorwith a metallized dome.

FIG. 1 shows the cross-section of a reflector according to the inventionin a plane passing through its axis of symmetry T. The reflectorcomprises a sphere 1 of dielectric material and a metal belt 2 placed ata distance from the centre of the sphere equal to p, this length beingstandardized by taking the external radius of the sphere 1 as the unitof length. The planes perpendicular to the axis of symmetry T andpassing through the edges of the belt are designated respectively as theupper plane and the lower plane of the belt.

The figures and calculations are given for the case where the belt issymmetrical with respect to an equatorial plane, the edge of which isindicated by the line U in FIG. 1. The half opening-angle of the belt isdesignated by 0; this angle is shown in FIG. l by the acute angle madewith the plane U by a straight line passing through the centre of thesphere and the edge of the belt.

As has -been previously stated, the law of variation of the permittivitywith the distance from the centre of the sphere must be such that theimage of the point at infinity is located practically at a distance pfrom the centre. Amongst the various possible laws which enable thiscondition to be obtained, there has been chosen for the followingdescription the law indicated by A. S. Gutman in the article entitled,Modified Luneberg Lens, in the review, Journal of Applied Physics, forJuly 1954, pages 855 to 859, and which is expressed by:

the variation of permittivity is not continuous, but the retiector isformed by the superimposition of homogeneous spherical layers, as shownin FIG. 2, the permittivity of each layer at the half-thickness being inthe neighbourhood of that given by Formula l and the number of layers issuflcient for the calculations which follow to be applicable in practiceto this preferred form of embodiment.

As the reector is symmetrically spherical, the paths of the rays areplanar curves; they are arcs of an ellipse having as their centre thecentre O of the sphere. One of these paths is illustrated in FIG. 3. Theplanar wave meets the outer surface of the sphere at B. In this sphere,it passes over the arc BCA, where C is the point at which it meets thespherical surface which carries the belt and A is the image point of thepoint at infinity. The equation of this curve referred to Cartesiancoordinates with an origin is:

xLZxy cotg at{-y2[cotg2 ati-p2(l{cotg2 a)]-p2=0 (2) in which theparameter a in the angle made by the tangent to the trajectory at thepoint A with the straight line OA. As the straight line OB which joinsthe centre O to the point B at which the trajectory passes out of thereflector is conjugated with OA, the angle at O of the straight line OBwith the outline of the plane V is equal t0 a.

A relation which subsequently be useful is that which associates theangle a with the angle ,b made by the plane V with the straight line OCjoining the centre O to the point C. In order to find this relation, xand y are replaced in Equation 2 by their following values:

In order to show the advantages of the device according to the presentinvention, the useful surface of the reflector according to theinvention will be determined and compared with that of retiectors withan external belt.

In FIG. 4 there is shown a cross-section of the reliector made in aplane passing through the centre of the sphere and perpendicular to thedirection of the incident rays, and on which there has been drawn theprojection of the edges of the front portion of the belt and at B theprojection of the point at which an incident ray meets the outer sphere.FIG. 5 shows a prole view in cross-section in a plane passing throughthe centre of the sphere and perpendicular to the preceding plane. Thebelt is indicated at 2, as in FIG. l.

As has been previously stated, the belt may be inclined with respect tothe reference plane. An inclination about the axis Y parallel to theincident rays obviously has no inuence on the useful surface, and willnot be considered in the present description. There will thus beconsidered only the case Where the belt is inclined about the axis X atright angles to the incident rays and located in the reference plane.The angle of inclination of the belt Z with respect to the referenceplane having its edge at line Y on FIG. 5 is designated by p.

It is proposed to determine the form and the magnitude of the usefulsurface of the reflector, that is to say the surface measured in theplane of FIG. 4 and in which the rays can be sent back to the sourceafter a single reection, without being stopped by the belt. The saiduseful surface is composed of two portions symmetrical with respect tothe axis X, and the present study can thus be limited to the upperportion.

The useful surface is limited by the rays which touch the edge of thebelt. The limiting curve of the surface is thus the projection on theplane of FIG. 4 of the line which forms the locus of intersection of therays passing over the edge of the belt with the outer surface of thereflector.

This line will be determined by cutting the plane of FIG. 4 by a planehaving its edge represented by line Z on the plane of the drawing, andwhich makes an angle 1 with the axis X. The plane Z cuts the upper planeof the belt and the sphere of radius p at a point C. The upper plane ofthe belt is shown in FIG. 5 by its edge FG; it

meets the axis Y at a point P such that:

sin 0 P0 p Bin go (6) and P01=p sin 0 cot-g ip (7) The perpendicular atO to the axis Y cuts the straight line PF at a point S1, such that:

sin 6 The upper plane of the belt is shown in FIG. 6. It cuts the beltalong a circumference having a centre O1 and a radius p cos 0. The edgeof the plane Z on the plane of FIG. 6 is a straight line PC forming withP01 an angle such that:

tg=sin ga cotg 11 (9) The perpendicular at C to the axis P01 meets thisstraight line at C1. The point S1 defined on FIG. 5 is referred at S1 onFIG. 6. E is the foot of the perpendicular dropped from O1 on PC.

It is possible to calculate the length of S1C1, which is usefulsubsequently. It is equal to:

Replacing PS1 by its value given in Equation 8, PE and EC by theirvalues deduced from the relations in the triangles of FIG. 6, and takingacount of (7), there is obtained:

11 above is equal to the length S1C1 of FIG. 5; from this there isobtained:

However, it is now possible to consider FIG. 3 as representing thesection of the reector by the plane perpendicular to the plane of FIG.4, represented by Z. FIG. 3 thus shows the trajectory of a ray in theplane Z. From this gure it can be seen that:

cos

Replacing OD by its expression in Equation l2 and substituting in thisequation the value of S1C1 obtained from (11), there is obtained afterseveral conversions:

oos|1b= L -[cos ,asin fn/cos2 -oos2 ecos2 1; -sin @sin e l-cos2 17 cos2e inclined and maximum inclination.

In the rst case, go=0, and the Equation 14 becomes:

2 2 cos=x cos cos 1;

sin 1; In the second case, p=, and Equation 14 becomes:

cos2 0 sin2 1,-sn2 0 c s o y l-cos2 n cos2 0 (16) of example, there havebeen shown in FIGS. 7, 8 and 9 determined for 0:25, FIGS. 7, 8 and 9corresponding respectively to go=0, p=15 and p=25. The curve in brokenlines corresponds to p=0.6, the curve in chain-dotted lines to p=0.8 andthe curve in full lines the case where the belt is lo- The advantagesindicated above become still more marked as p becomes smaller. On theother hand however, when pis smaller, the reflector is heavier sinceEquation 1 results in higher permittivities.

I claim:

1. A spherical reflector for high-frequency electro-magnetic wavescomprising a sphere of low-loss dielectric belt to said centre, theradius of the sphere being .taken as the unit of length.

3. A reiector between 0.50 and 0.90.

4 A spherical reflector for high-frequency electromag 5. A reflector forhigh-frequency electro-magnetic waves as claimed in claim 4, in whichthe in each of the homogeneous electric material is given byelectro-magnetic 1n which the distance from lthe belt to the centre ofsaid sphere, constituted by concentric homogeneous layers of low-lossdielectric material and the radius of said sphere are in a ratiocomprised between 0.50 and 0.90.

References Cited bythe Examiner HERMAN KARL SAALBACH, Primary Examiner.

1. A SPHERICAL REFLECTOR FOR HIGH-FREQUENCY ELECTRO-MAGNETIC WAVES COMPRISING A SPHERE OF LOW-LOSS DIELECTRIC MATERIAL, THE PERMITTIVITY OR DIELECTRIC CONSTANT OF WHICH IS THE SAME AT ALL POINTS LOCATED AT THE SAME DISTANCE FROM THE CENTRE OF SAID SPHERE AND VARIES IN ACCORDANCE WITH A LAW SUCH THAT THE IMAGE OF A POINT AT INFINITY IS FOMED PARCTICALLY IN A POINT LOCATED ON A SHPERICAL SURFACE CONCENTRIC WITH AND WITHIN SAID SPHERE, THE RADIUS OF SAID SURFACE BEING LESS THAN THAT OF SAID SPHERE; AND A REFLECTING BELT OPAQUE TO SAID WAVES, AND HAVING THE FORM OF A SPHERICAL ZONE CONCENTRIC WITH SAID SPHERE AND CONSTITUTING A PORTION OF SAID SPHERICAL SURFACE, SAID SPHERICAL ZONE BEING LIMITED BY TWO LINES PARALLEL TO AN EQUATORIAL PLANE OF THE SPHERE AND LOCATED ON EACH SIDE OF THIS PLANE WHEREBY FOR EFFECTIVE REFLECTOR ACTION THE DISTANCE BETWEEN SAID TWO PARALLEL LINES LIMITING SAID SPHERICAL ZONE IS LESS THAN THE CORRESPONDING DISTANCE WOULD BE WITH SAID SPHERICAL SURFACE COINCIDINT WITH SAID SPHERE. 